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Register a new userDyson models under renormalization and in weak fields
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Aernout Coert Daniel van Enter
Publication date
2017
fields
Physics
and research's language is
English
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We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form $frac{1}{|i-j|^{alpha}}$ mainly focusing on the range of slow decays $1 < alpha leq 2$. We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as $(|i|+1)^{- gamma}$. We study how the critical decay power of the field, $gamma$, for which the phase transition persists and the decay power $alpha$ of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.
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We consider long-range self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)asymp|x|^{-d-alpha}$ with $alpha>0$. The upper-critical dimension $d_{mathrm{c}}$ is $2(alphawedge2)$ for self-avoiding walk and the Ising model, and $3(alphawedge2)$ for percolation. Let $alpha e2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{alphawedge2-d}$, where the constant $Cin(0,infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $alpha<2$ and $alpha>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.
We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of the zeros. In particular, we show that for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the $q$-state Potts model for $q$ large enough.
In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $Delta<1$. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $a=b=1$ and $cge1$, and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.
This is a short review of the two papers on the $x$-space asymptotics of the critical two-point function $G_{p_c}(x)$ for the long-range models of self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$, defined by the translation-invariant power-law step-distribution/coupling $D(x)propto|x|^{-d-alpha}$ for some $alpha>0$. Let $S_1(x)$ be the random-walk Green function generated by $D$. We have shown that $bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($alpha>2$) to Riesz ($alpha<2$), with log correction at $alpha=2$; $bullet~~G_{p_c}(x)simfrac{A}{p_c}S_1(x)$ as $|x|toinfty$ in dimensions higher than (or equal to, if $alpha=2$) the upper critical dimension $d_c$ (with sufficiently large spread-out parameter $L$). The model-dependent $A$ and $d_c$ exhibit crossover at $alpha=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functions (with log corrections, if $alpha=2$) to optimally control the lace-expansion coefficients $pi_p^{(n)}$, and (iii) probabilistic interpretation (valid only when $alphale2$) of the convolution of $D$ and a function $varPi_p$ of the alternating series $sum_{n=0}^infty(-1)^npi_p^{(n)}$. We outline the proof, emphasizing the above key elements for percolation in particular.
Using the results on the $1/n$-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with $n$ varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.
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